Remark

is the free (∞,1)-cocompletion of XX, hence the free completion under (∞,1)-colimits. Under the interpretation of colimits as sums, this means that it is analogous to the vector spaces spanned by the basisCC.

Accordingly an arbitrary locally presentable (∞,1)(\infty,1)-category is analogous in this sense to a sub-space of a vector space spanned by a basis.

Function spaces

Notice that for each objectX∈HX \in \mathbf{H} the over-(∞,1)-toposH/X\mathbf{H}/X is the little topos of (∞,1)(\infty,1)-sheaves on XX. So to the extent that we think of these as function objects , and of locally presentable (∞,1)(\infty,1)-categories as linear spaces, we may think of H/X\mathbf{H}/X as the ∞\infty-vector space of ∞\infty-functions on XX

There is a further right adjointv*v_*. For the present purpose the relevance of its existence is that it implies that both v!v_! as well as v*v^* are left adjoints and hence both preserve (∞,1)-colimits. Therefore these are morphism in Pr(∞,1)Cat and hence behave like linear maps on our function spaces H/X\mathbf{H}/X and H/Y\mathbf{H}/Y.

When we think of base change in the context of linear algebra on sheaves, we shall write ∫X/Y:=v!\int_{X/Y} := v_!

The objects on the right we may again think of as (∞,1)(\infty,1)-profunctors X⇸YX &#x21F8; Y. So in particular the kernel (A→X×Y)∈∞Grpd/(X×Y)(A \to X \times Y) \in \infty Grpd/(X \times Y) is under this equivalence on the right hand identified with an (∞,1)(\infty,1)-profunctor